Is Euclid dead? or Should Euclidean geometry be taught to high school students?

Question

Apparently Euclid died about 2,300 years ago (actually 2,288 to be more precise), but the title of the question refers to the rallying cry of Dieudonné, "A bas Euclide! Mort aux triangles!" (see King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry by Siobhan Roberts, p. 157), often associated in the popular mind with Bourbaki's general stance on rigorous, formalized mathematics (eschewing pictorial representations, etc.). See Dieudonné's address at the Royaumont seminar for his own articulated stance.

In brief, the suggestion was to replace Euclidean Geometry (EG) in the secondary school curriculum with more modern mathematical areas, as for example Set Theory, Abstract Algebra and (soft) Analysis. These ideas were influential, and Euclidean Geometry was gradually demoted in French secondary school education. Not totally abolished though: it is still a part of the syllabus, but without the difficult and interesting proofs and the axiomatic foundation. Analogous demotion/abolition of EG took place in most European countries during the 70s and 80s, especially in the Western European ones. (An exception is Russia!) And together with EG there was a gradual disappearance of mathematical proofs from the high school syllabus, in most European countries; the trouble being (as I understand it) that most of the proofs and notions of modern mathematical areas which replaced EG either required maturity or were not sufficiently interesting to students, and gradually most of such proofs were abandoned. About ten years later, there were general calls that geometry return, as the introduction of the alternative mathematical areas did not produce the desired results. Thus EG came back, but not in its original form.

I teach in a University (not a high school), and we keep introducing new introductory courses, for math majors, as our new students do not know what a proof is. [Cf. the rise of university courses in the US that come under the heading "Introduction to Mathematical Proofs" and the like.]

I am interested in hearing arguments both FOR and AGAINST the return of EG to high school curricula. Some related questions: is it necessary for high-school students to be exposed to proofs? If so, is there is a more efficient mathematical subject, for high school students, in order to learn what is a theorem, an axiom and a proof?

Full disclosure: currently I am leading a campaign for the return of EG to the syllabus of the high schools of my country (Cyprus). However, I am genuinely interested in hearing arguments both pro and con.

Note. This question was also asked in mathoverflow.

Answer 1

I'd like to tackle the question from another point of view than JPBurkes answer: If you accept, that mathematical argumentation (whatever level) is an essential part of mathematics courses in K-12, than Euclidean Geometry is a great way to implement this:

Visuality Euclidean Geometry deals with objects that can be easily visualized. It can be properly served on all three represantative layers: enactive, iconic, symbolic.

Scaling of Argumentation level Theorems in Euclidean Geometry can be proven or argumented for on different argumentation levels: intuitively formal-rigorous, abstractedly formal-rigorous (Euclid's way), with generalizable examples, using intuitive knowledge (symmetry, movement invariance, …).

Loss of calculations Many proofs in Euclidean Geometry include no calculations at all, others only as small substeps. Consequently, students can learn that way, that maths is not just calculations. This sounds trivial, but it is often a problem: They hesitate in Euclidean Geometry, because they cannot simply calculate a result. The result isn't even a number or a value, but the theorem itself.

Answer 2

I am interested in hearing arguments both FOR and AGAINST the return of EG to high school curricula. Some related questions: is it necessary for high-school students to be exposed to proofs?

I am going to address part of this question, specifically the changing role of the notion of proof as it appears in standards (influenced by research).

My brief response borrows heavily from the document I am referencing (Yackel & Hanna, 2003); by responding I intend to connect your question to something underlying the inclusion of mathematical practices (and/or mathematical processes) in recent standards documents here in the USA. This inclusion is based on mathematics education theory. I do not address arguments for including Euclidean geometry and geometric proofs as curriculum content. As proof and proving are not my particular area, I am hoping that what I can provide here points you towards research and arguments that help you address the part of your question that is tied to "proofs."

Many math ed researchers tightly associate learning and reasoning (see a quote from Thompson in Yackel & Hanna, 2003, p. 227). Von Glasersfeld (the prominent theorist) "posited that knowledge is built up by the cognizing individual (2003, p. 227)."

The importance placed on mathematical reasoning in the learning process has been embraced by some recent research-based approaches to math education, Being able to communicate how you know something now becomes an inherent part of mathematics instruction (and classroom activity for both teachers and students). Therefore, at all levels, argumentation and justification become a practice that must be suffused throughout the curriculum rather than a specific content target for a particular grade band. This is not only true for the NCTM Principles and Standards for School Mathematics (National Council of Teachers of Mathematics, 2000), but also for the recent Common Core State Standards with their section on mathematical practices.

You will find some research-based justifications for this approach to standards-based education in the Yackel and Hanna (2003) chapter. A more recent book (Stylianou, Blanton, & Knuth, 2009) also provides insights into how research informs the inclusion of proof throughout the grades, and as early as kindergarten.

I hope this acts as a decent starting point to sources that argue strongly for proof as a mathematical practice, which would provide a foundation for secondary-level proof and proving that might be more recognizable in traditional curricula. Yackel and Hanna (2003) note the difficulty some students face in secondary school when they have not had an underlying mathematical education that includes justification and argumentation (p. 234).

In short, we can see a research-supported change in the way proof and proving is approached in K-12. We see the idea that argumentation and justification are practices students need throughout their mathematics education rather than being introduced to "proofs" as an activity late in their public school education. Additionally, that these practices support students in learning "proof" later on.

The chapters in Stylianou, Blanton and Knuth (2009) make the argument for the broadening of the notion of proof, and are a much more detailed look at how and why researchers are seeing the need for this perspective change. This doesn't argue against Euclidean proof, but I think it does argue against that being the first introduction to the idea of proving, and of the practice of justification as an inherent part of knowing in mathematics.

ADDENDUM:

I checked, and there is proof used to "refute or support conjectures" in the 9-12 Geometry Standard of the NCTM Principles and Standards for School Mathematics. (Link is behind a paywall)

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Cited

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA.

Stylianou, D. A., Blanton, M. L., & Knuth, E. J. (2009). Teaching and learning proof across the grades: A K-16 perspective. Routledge. Retrieved from http://books.google.com/books?hl=en&lr=&id=8RCRAgAAQBAJ&oi=fnd&pg=PP1&dq=proof+blanton&ots=c5wYvMHgOT&sig=-5-wM5dmwLG_ZWCJeLfIpyEUDpA

Yackel, E., & Hanna, G. (2003). Reasoning and proof. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.), A research companion to principles and standards for school mathematics (pp. 227–236). Reston, VA: National Council of Teachers of Mathematics.

Answer 3

I teach at the college level in the US, where geometry is a standard part of the high school curriculum, so my students are all supposed to have had previous exposure to it. However, probably only a certain percentage (50%?) of students here take a geometry course that requires them to write proofs. I assume that in many cases in the US, the high school geometry teacher simply isn't willing to grade a stack of proofs from five classes, each with an enrollment of 40.

What frightens me is the results I get when I ask students to write even a trivial proof. For example, I introduce the vector cross product and give both its characterization in terms of components and its geometrical characterization. I then assign a homework problem in which the student is asked to prove that $\mathbf{A}\times(k\mathbf{B})=k(\mathbf{A}\times\mathbf{B})$. The most common response (among students who write an answer at all) is to make up an example and demonstrate that the property holds for that example.

If students think this is a valid mode of reasoning, then I suppose they believe that all presidents of the United States are African-American, since we have an example that proves it. This is not a defect in their mathematical preparation, it's a defect in their critical thinking skills that could perhaps have been addressed through their mathematical training.

These are the results in an educational system that does, at least in theory, require proof-writing in high school. I shudder to imagine how much worse it would be in a system where there had not even been any attempt to teach proof-writing. When Abraham Lincoln was a country lawyer in Illinois, traveling on horseback to county seats and sleeping in boarding houses, he always carried a copy of Euclid with him, which he would study at night by lamplight as a model of logical argumentation. I wonder how Lincoln would have turned out if he'd never been exposed to Euclid.

Answer 4

Since no one here seems to be speaking against the teaching of Euclidean Geometry in highschool I guess I'll explain a few structural problems I have with EG.

My main complaint is what Toscho sees as an good feature. In particular:

Loss of Calculations

Ok, fair enough, standard EG following in the steps of the long dead Euclid proceeds from Axioms to Theorem with out so much as a number in sight. I studied it in high school, admittedly the construction of perpendicular bisectors etc. by compass and straight-edge alone is fascinating. But, I ask, is this loss of calculation natural in our modern age ? Have students already been exposed to math which is over a millenia beyond Euclid before the take EG ? The answer to my question is quite obvious since about the 17th century. From a student's perspective, what I think is a more accurate portrait of EG is this:

Artificial loss of calculations

Ok, so, I can't trisect an angle. Really, because if I use a ruler and a compass we can trisect an angle. Or, to be more direct, if I have a protractor and the idea of dividing real numbers by $3$ then trisecting an angle is not a big deal. In short, students already know the logical successor of EG, in particular analytic geometry is woven throughout our modern discussion of Mathematics and its application to geometry.

I would argue, students would be better served by studying proofs which involve calculation. One of the great absurdities, created in part by the existence of the "transition to proofs" class, is this idea that there is some conceptual divide between math involving equations and math involving proofs. Certainly not. In fact, what we have is merely a dichotomy in standards. On the one hand, the student has an algebra course where to find the vertex of a parabola they simply use an equation, often with no expectation of supporting argumentation or explanation, just the plug-and-chug method so to speak. Then that same student in their EG course is asked to prove something which is frankly obvious with a 5 line column proof where they have to refer to each Axiom and step with more precision than we require in some proof classes at the undergraduate level.

I can't say what is best to replace EG with, but, I would tend to agree with Burke that is not actually about curriculum. The real struggle here is about standards. Why do we insist on having standards in EG, but, not so much in other areas of math. Students should be expected to explain why they are calculating in a particular way. The over-emphasis of problem-solving verses the coherence of theory is the true root of the problem.

I can't blame teachers most places, there is simply no intellectual will-power or structural room to build good mathematics. We're too busy meeting SOL's and worrying about inane federal guidelines and such. The idea that students should be held to a higher standard is at odds with every other societal direction, so, I don't think there is a global fix. The solution is at most local, with particular educators investing their vision and coherent understanding of Math where proof and critical analytical thinking are woven into every math class, not just the EG course. This is why I am a strong advocate for every math teacher having at least an undergraduate degree in math.

Answer 5

One difficult issue with Euclidean geometry is that Euclid's approach, although it was an important step historically, is highly non-rigorous by modern standards. This is especially clear in the definitions and basic notions (what does "a line is breadthless length" mean?). But there's a much bigger problem that already comes up in Prop 1, which is that there's no way to see from the axioms that those two circles intersect at all! Indeed $\mathbb{Q}^2$ satisfies all of Euclid's axioms, but the circles won't intersect since they share no rational points. As Hilbert and others explained, there are ways to fix this issue by adding a lot more axioms. But at that point you're almost doing analysis anyway.

I think number theory, group theory, analysis and graph theory are all better ways to introduce proofs than Euclidean geometry.

Answer 6

I am generally favorable to Euclidean geometry and proofs, within limits, but we do need to understand why learning proofs is so difficult that many countries have reduced such content considerably. Only by appreciating why learning proofs in high school is problematic can we hope to overcome the difficulties.

A considerable amount of research, especially in the 1980s, has shown that students are not as successful with geometric proofs as they seem. For example, in the US, where proofs are still usually taught, teachers developed a system of "two-column" proofs to make the process easier to grade and more "rigorous". Unfortunately two-column proofs often very predictably go through similar processes which teachers manage to teach in such a way that many students learn to successfully imitate them without any deep understanding of what they are actually doing. For example, a great many proofs students learn to do can be solved with a two-column proof by identifying the given information in a triangle, identifying the appropriate theorem (ASA, SAS, SSS, etc.) and continuing to the conclusion. The whole question of why we are doing this in the first place, or what it all means, is given little emphasis.

One large study (Usiskin, Z. [1982]. Van Hiele levels and achievement in secondary school geometry. Chicago, IL: University of Chicago) found that for many, perhaps most, students who had completed a full year of EG, not only did they not understand what they were doing, but they did not yet even have the prerequisites to begin a study of geometric proofs. There has been a considerable attempt to fill in the gaps found in these studies (both in geometric content and logical reasoning) in earlier grades, but it is not still clear if today's students are ready to fully appreciate EG, especially when taught all at once in a one-year course, as is done in the US.

There have been similar findings in Europe (e.g. Gutiérrez, Á., & Jaime, A. [1998]. On the assessment of the Van Hiele levels of reasoning. Focus on Learning Problems in Mathematics, 20(2/3), 27-46.)

Answer 7

I am a high school math teacher and whenever I teach the geometry class, I always do full proofs. I use the geometry class to show kids how we construct all the ideas of geometry from a single point and I use proofs to explain to them the idea of mathematical truths. I teach at an IB school so this preps them for the TOK course they'll take in a few years. But proofs in the Geometry unit really gives me the opportunity to wax philosophical and some the kids from countries where they are just expected to regurgitate facts really appreciate learning that there is an answer "why is this true?" beyond "because I said so.", behind everything they have learned.

Answer 8

When I went to high school in the early 80s, I studied geometry and did formal mathematical proofs. I then studied economics undergrad and business grad school. When I decided to return in my 30s to school to pursue a master's degree in mathematics, I did not need an undergraduate math degree to apply to the graduate programs that interested me, but I did need to have completed certain courses (like Elementary Number Theory). When I started taking these upper division math classes, some required that I write proofs. Without my high school experience, I would have been lost. Further, whether proofs are introduced in geometry or in some other way in hgih school, learning and practicing logical thinking is foundational to a number of undergraduate disciplines. Thus proofs seem to be an essential college preparatory skill.

Answer 9

I certainly appreciate the planar Euclidean geometry but my personal inclination is rather to teach combinatorics, graph theory, and number theory, which is exactly what I'm doing when I teach university math classes to future school teachers.

The advantages I see in those are that you can get your proofs in an incremental way: you can experiment with small numbers and pictures before trying to discern the general formula or statement. It is also easier to refute the wrong proof by presenting an example where a certain step fails (not that it is easier to create a counterexample, but it is easier to convince the student that it is a counterexample). It is also easier (for me) to create chains of problems (say, counting ones) of gradually increasing difficulty, while in the Euclidean geometry you need to be quite inventive not to jump from "trivialities" to "hopelessly difficult" problems. As to computer visualization, the integer arithmetic seems more reliable to me than the graphic: if the machine tells me that 239+456=695, I generally believe it, but if I see two lines and a circle passing through one point, I'm always tempted to look around in search of a magnifying glass.

In general, I do not care much which subject a proof course is based upon as long as it is a proof course and not the one about memorization of formulae and algorithms. Unfortunately, it is quite easy to turn the former into the latter, and then we do not need students for such courses: computers are easier to program and the results of such programming are more reliable. The only advantage of a human over a machine nowadays is that a human can think illogically to produce logical derivations and to build working contraptions and, while it is still unclear how exactly this process works, it is pretty clear that one needs practice to get good at it and that the proof courses are the best tool so far we have to develop this skill.

Answer 10

"Should Euclidean geometry be taught to highschool students?" Certainly, yes. The whole idea that it shouldn't is a misguided mistake with identifiable historical sources and based on identifiable philosophical errors.

Searching through both the question and the existing answers I notice that there has been no mention made until now of the so-called New Math. I think this should be mentioned is because the basic impetus behind Bourbaki, Dieudonne "death to triangles" call, and New Math is the same, the philosophical errors are the same, and the reasons for failure are the same. For a detailed analysis of the history of the New Math educational debacle see this publication.

More specifically, the philosophical errors involved are traceable to Piaget. Piaget was interested in analyzing the structures of the learner's mind. Piaget claimed an affinity between such structures and the structures of "modern" mathematics. Such an identification was a non-sequitur but it influenced people like Begle who were in key positions in the US to affect pre-university education.

Answer 11

Yes Euclid is deeply important. Its important for a few reasons:

1. Its visual and therefore leverages preexisting intuitions about space that people have.

2. It the easiest intro to axiomatic mathematics which forms the basis for all future math in university math depts

3. Its doubles as a science...everything in Euclidean geometry can actually be tested in the real world.

4. It is also the basis for a lot of the philosophical tradition and epistemological tradition of the west.

5. There is rigor and precision to how Euclid proceeds that I want in the heads of people as an example to strive for. I want them to desire this and look for it. If you surround yourself with ugliness you won't know to desire beauty because you will never see a good example of it. Euclid is did for truth what the Sistine chapel did for beautiful churches

6. It is constructive and synthetic. Current math is mostly analytic. A lot of math today is also non-constructive. In the mathematical tradition Euclid provides a deeply important aesthetic thread and I don't want it to be lost!!

7. It encourages the right epistemological morals. People complain that Euclid artificially constrains himself. But they don't appear to understand why. There is an important a lesson to be had here. The Greeks realized how easily one could be lead astray. By constraining what was assumed to as few as possible assumptions they hoped not to be led astray. This kind of fear error, limitation of assumptions and stress on using deductive proofs as far as possible is incredibly important. Newton understood it which is why he wouldn't use calculus and proved all his results geometrically.

8. It has inspired countless generations of mathematicians, scientists and philosophers Consider Russell's reaction:

At the age of eleven, I began Euclid, with my brother as tutor. This was one of the great events of my life, as dazzling as first love. I had not imagined there was anything so delicious in the world.

or the brilliant Israeli logician Saharon Shelah:

But when I reached the ninth grade I began studying geometry and my eyes opened to that beauty—a system of demonstration and theorems based on a very small number of axioms which impressed me and captivated me.

It difficult to fully express all that is going on in Euclid but it needs to be internalized into the heads of scientists and mathematicians. My fear is that it hasn't been in this generation and we will suffer the consequences in junk science and amorphous fuzzy results for decades to come.

The greatest reason to study Euclid is therefore for me that it points away from the inexorable entropic forces of motivated reasoning, advocacy science and pure bullshit that are Western scientific culture is currently succumbing to. And towards something much much better.

Answer 12

While talking about proofs, I notice that nobody is mentioning the conditions: while performing mathematics you can't just do what you want: all conditions need to be fulfilled before you can actually do something.

When explaining a proof to a class, it's very important that the teacher mentions the conditions, and while writing the proof, the teacher should clearly indicate at what point the mentioned conditions are used (e.g. while proving Pythagoras' theorem, the teacher should clearly indicate why this only applies to right-angle triangles).

This way of working learns the students why the conditions are so important and automatically they will watch their steps while setting up complicated reasonings (not only in math, by the way).

Answer 13

This question has just appeared in the list of currently active questions, so I thought to add my penny's worth.

I have been tutoring a struggling mathematics student (mid-teens) by spending an hour a week with him, helping him understand the material he is having trouble with.

He showed me what he was doing in geometry. Apparently they have been given an arbitrary bunch of theorems to learn, one of which being the Inscribed Angle Theorem. (In fact, they don't even have to "learn" them, they are provided a sheet of theorems which they may refer to during the course of any tests they are doing.) None of these theorems has even been proved in the classroom. They are just taught to be able to use them. As a result, my student feels a little glum, because he believes that geometry is way over his head.

So I drew a few lines on the standard diagram, pointed out the isosceles nature of the triangles formed as a result (obvious once you've had it pointed out to you) and encouraged him to use what he knew about the number of degrees that the angles of a triangle add up to, remind him what algebra was for, and watched him as he proved that indeed, the angle subtended by a chord is exactly twice at the centre what it is when the angle is on the perimeter.

He was right indignant. "That was really simple! Why couldn't the teacher have showed us that?" So the next time I saw him, I left my copy of Euclid for him to look at. He loves it.